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- 1. Appendix C: Advanced Relational Database Design s Reasoning with MVDs s Higher normal forms q Join dependencies and PJNF q DKNFDatabase System Concepts, 1 st Ed. C.1 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 2. Theory of Multivalued Dependencies s Let D denote a set of functional and multivalued dependencies. The closure D+ of D is the set of all functional and multivalued dependencies logically implied by D. s Sound and complete inference rules for functional and multivalued dependencies: 1. Reflexivity rule. If α is a set of attributes and β ⊆ α, then α →β holds. 2. Augmentation rule. If α → β holds and γ is a set of attributes, then γ α→γ β holds. 3. Transitivity rule. If α → β holds and γ α→γ β holds, then α →γ holds.Database System Concepts, 1 st Ed. C.2 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 3. Theory of Multivalued Dependencies (Cont.) 4. Complementation rule. If α β holds, then α R–β–α holds. 5. Multivalued augmentation rule. If α β holds and γ ⊆ R and δ ⊆ γ, then γ α δ β holds. 6. Multivalued transitivity rule. If α β holds and β γ holds, then α γ – β holds. 7. Replication rule. If α β holds, then α β. 8. Coalescence rule. If α β holds and γ ⊆ β and there is a δ such that δ ⊆ R and δ ∩ β = ∅ and δ γ, then α γ holds.Database System Concepts, 1 st Ed. C.3 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 4. Simplification of the Computation of D+ s We can simplify the computation of the closure of D by using the following rules (proved using rules 1-8). q Multivalued union rule. If α β holds and α γ holds, then α βγ holds. q Intersection rule. If α β holds and α γ holds, then α β∩γ holds. q Difference rule. If If α β holds and α γ holds, then α β–γ holds and α γ – β holds.Database System Concepts, 1 st Ed. C.4 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 5. Example s R = (A, B, C, G, H, I) D = {A B B HI CG H} s Some members of D+: q A CGHI. Since A B, the complementation rule (4) implies that A R – B – A. Since R – B – A = CGHI, so A CGHI. q A HI. Since A B and B HI, the multivalued transitivity rule (6) implies that B HI – B. Since HI – B = HI, A HI.Database System Concepts, 1 st Ed. C.5 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 6. Example (Cont.) s Some members of D+ (cont.): q B H. Apply the coalescence rule (8); B HI holds. Since H ⊆ HI and CG H and CG ∩ HI = Ø the , coalescence rule is satisfied with α being B, β being HI, δ being CG, and γ being H. We conclude that B H. q A CG. A CGHI and A HI. By the difference rule, A CGHI – HI. Since CGHI – HI = CG, A CG.Database System Concepts, 1 st Ed. C.6 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 7. Normalization Using Join Dependencies s Join dependencies constrain the set of legal relations over a schema R to those relations for which a given decomposition is a lossless-join decomposition. s Let R be a relation schema and R1 , R2 ,..., Rn be a decomposition of R. If R = R1 ∪ R2 ∪ … ∪ Rn, we say that a relation r(R) satisfies the join dependency . *(R1 , R2 ,..., Rn) if: r =∏R1 (r) ⋈ ∏R2 (r) ⋈ …… ⋈ ∏Rn(r) A join dependency is trivial if one of the Ri is R itself. s A join dependency *(R1, R2) is equivalent to the multivalued dependency R1 ∩ R2 R2. Conversely, α β is equivalent to *(α ∪(R - β), α ∪ β) s However, there are join dependencies that are not equivalent to any multivalued dependency.Database System Concepts, 1 st Ed. C.7 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 8. Project-Join Normal Form (PJNF) s A relation schema R is in PJNF with respect to a set D of functional, multivalued, and join dependencies if for all join dependencies in D+ of the form *(R1 , R2 ,..., Rn ) where each Ri ⊆ R and R =R1∪ R2 ∪ ... ∪ Rn at least one of the following holds: q *(R1 , R2 ,..., Rn ) is a trivial join dependency. q Every Ri is a superkey for R. s Since every multivalued dependency is also a join dependency, every PJNF schema is also in 4NF.Database System Concepts, 1 st Ed. C.8 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 9. Example s Consider Loan-info-schema = (branch-name, customer-name, loan-number, amount). s Each loan has one or more customers, is in one or more branches and has a loan amount; these relationships are independent, hence we have the join dependency s *(=(loan-number, branch-name), (loan-number, customer-name), (loan- number, amount)) s Loan-info-schema is not in PJNF with respect to the set of dependencies containing the above join dependency. To put Loan-info-schema into PJNF, we must decompose it into the three schemas specified by the join dependency: q (loan-number, branch-name) q (loan-number, customer-name) q (loan-number, amount)Database System Concepts, 1 st Ed. C.9 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 10. Domain-Key Normal Form (DKNY) s Domain declaration. Let A be an attribute, and let dom be a set of values. The domain declaration A ⊆ dom requires that the A value of all tuples be values in dom. s Key declaration. Let R be a relation schema with K ⊆ R. The key declaration key (K) requires that K be a superkey for schema R (K → R). All key declarations are functional dependencies but not all functional dependencies are key declarations. s General constraint. A general constraint is a predicate on the set of all relations on a given schema. s Let D be a set of domain constraints and let K be a set of key constraints for a relation schema R. Let G denote the general constraints for R. Schema R is in DKNF if D ∪ K logically imply G.Database System Concepts, 1 st Ed. C.10 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 11. Example s Accounts whose account-number begins with the digit 9 are special high-interest accounts with a minimum balance of 2500. s General constraint: ``If the first digit of t [account-number] is 9, then t [balance] ≥ 2500. s DKNF design: Regular-acct-schema = (branch-name, account-number, balance) Special-acct-schema = (branch-name, account-number, balance) s Domain constraints for {Special-acct-schema} require that for each account: q The account number begins with 9. q The balance is greater than 2500.Database System Concepts, 1 st Ed. C.11 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100
- 12. DKNF rephrasing of PJNF Definition s Let R = (A1 , A2 ,..., An) be a relation schema. Let dom(Ai ) denote the domain of attribute Ai, and let all these domains be infinite. Then all domain constraints D are of the form Ai ⊆ dom (Ai ). s Let the general constraints be a set G of functional, multivalued, or join dependencies. If F is the set of functional dependencies in G, let the set K of key constraints be those nontrivial functional dependencies in F+ of the form α → R. s Schema R is in PJNF if and only if it is in DKNF with respect to D, K, and G.Database System Concepts, 1 st Ed. C.12 ©VNS InfoSolutions Private Limited, Varanasi(UP), India 22100

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